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Theorem fusgrvtxfi 29351
Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)
Hypothesis
Ref Expression
isfusgr.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
fusgrvtxfi (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Proof of Theorem fusgrvtxfi
StepHypRef Expression
1 isfusgr.v . . 3 𝑉 = (Vtx‘𝐺)
21isfusgr 29350 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
32simprbi 496 1 (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  Fincfn 8984  Vtxcvtx 29028  USGraphcusgr 29181  FinUSGraphcfusgr 29348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-fusgr 29349
This theorem is referenced by:  fusgrfupgrfs  29363  nbfusgrlevtxm1  29409  nbfusgrlevtxm2  29410  nbusgrvtxm1  29411  uvtxnm1nbgr  29436  cusgrm1rusgr  29615  wlksnfi  29937  fusgrhashclwwlkn  30108  clwwlkndivn  30109  fusgreghash2wsp  30367  numclwwlk3lem2  30413  numclwwlk4  30415
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